"I have even more responsibilities now!"
Translation:J'ai encore plus de responsabilités maintenant !
"J'ai même plus..." and "J'ai encore plus" have two different meanings in French. I don't think that distinction exists in English, both would be translated with "even more".
The distinction is the slightest, here is my try to explain it : "Même plus" would be used in a contradiction. For instance : "Bien sûr que je travaille en été, je travaille même plus que d'habitude !" -> if you thought I wasn't working during summer, you're wrong, I work even more!
"Encore plus" is used in a comparison. "Je lui ai déjà donné 200 euros, mais il en veut encore plus !" -> He wants more than the things I just gave him.
Maybe there is a way to translate "même plus" in English that I'm not aware of. I'd be grateful if you could tell me if there is. If there is not, both "encore" and "même" should be accepted for "even more".
Again it'd be a very slight nuance but maybe "encore plus" is more like "still more" in English and "meme plus" is analagous to "even more," the latter being a bit stronger: "the little boy ate 2 cookies but he'd still like more" vs. "No matter how much work you do that boss wants even more"